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Home » Solution: First, find the constant $ k $ using the given values: - Dachbleche24

Solution: First, find the constant $ k $ using the given values: - Dachbleche24

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 p^3 + q^3 = 5^3 - 3 \cdot 6 \cdot 5 = 125 - 90 = 35

📌 Question: Let $ a $, $ b $, and $ c $ be positive real numbers such that $ a + b + c = 1 $. Find the minimum value of $ rac{1}{a} + rac{1}{b} + rac{1}{c} $.

📌 Solution: We are to minimize $ rac{1}{a} + rac{1}{b} + rac{1}{c} $ under the constraint $ a + b + c = 1 $ with $ a, b, c > 0 $.

📌 By the AM-HM inequality:

📌 rac{a + b + c}{3} \geq rac{3}{rac{1}{a} + rac{1}{b} + rac{1}{c}} \Rightarrow rac{1}{3} \geq rac{3}{rac{1}{a} + rac{1}{b} + rac{1}{c}}

📌 Taking reciprocals (since all terms are positive):

📌 rac{1}{a} + rac{1}{b} + rac{1}{c} \geq 9

📌 Equality occurs when $ a = b = c = rac{1}{3} $. Thus, the minimum value is:

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